These things are trochoids of a sort - the trail left by a point on a wheel as it rolls around another wheel.

Normally trochoids are divided into two categories: Epitrochoids and hypotrochoids.

Epitrochoids are the trail from a wheel rolling around the outside of a fixed circle.

Hypotrochoids are those formed from a wheel rolling around the inside of a fixed circle.

These curves are from a more general class, with the circle being rolled around also turning.

The general equations for these are:

 x = size1*sin (q) + size2*sin(n*q-f)

y = size1*cos (q) + size2*cos(n*q+f))

where f depends on time.

By the way, if you watch carefully you should be able to see that all the rings in each of these animations are really the same.

 

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